Geometry Common CoreMidterm Review

Name: _______________________________ Date: __________ Period: ___________

1. In the diagram below, AEFB∥CGD and ¿ and GF are drawn.

If m∠EFG=32 ° and m∠ AEG=137 °, what is m∠EGF?

2. In the diagram below, congruent figures 1, 2, and 3 are drawn.

Which sequence of transformations maps figure 1 onto figure 2 and then figure 2 onto figure 3?

(1) a reflection followed by a translation

(2) a rotation followed by a translation

(3) a translation followed by a reflection

(4) a translation followed by a rotation

3. Which regular polygon has a minimum rotation of 45 ° to carry the polygon onto itself?

(1) octagon (3) hexagon

(2) decagon (4) pentagon

MIDTERM EXAM DATES

Wednesday, 1/15Thursday, 1/16

4. Which statement is sufficient evidence that ∆≝¿ is congruent to ∆ ABC?

(1) AB=DE and BC=EF

(2) ∠D≅∠A ,∠B≅∠E ,∠C≅∠F

(3) There is a sequence of rigid motions that maps AB onto DE, BC onto EF and AC onto DF .

(4) There is a sequence of rigid motions that maps point A onto point D, AB onto DE, and ∠B onto ∠E.

5. The coordinates of the vertices of ∆ RST are R (−2 ,−3 ) , S (8 ,2 ), and T (4 ,5 ). Which type of triangle is ∆ RST ?

(1) right (3) obtuse

(2) acute (4) equiangular

6. In the diagram to the right, a square is graphed in the coordinate plane. A reflection over which line does not carry the square onto itself?

(1) x=5 (3) y=x

(2) y=2 (4) x+ y=4

7. A sequence of transformations maps rectangle ABCD onto rectangle A' ' B' 'C ' 'D ' ' , as shown in the diagram below.

Which sequence of transformations maps ABCD onto A' B'C' D' and then mapsA'B'C' D' onto A' ' B' 'C ' 'D ' '?

(1) a reflection followed by a rotation

(2) a reflection followed by a translation

(3) a translation followed by a rotation

(4) a translation followed by a reflection

8. In the diagram below, which single transformation was used to map triangle A onto triangle B?

(1) line reflection (3) dilation

(2) rotation (4) translation

9. Two right triangles must be congruent if

(1) an acute angle in each triangle is congruent

(2) the lengths of the hypotenuses are equal

(3) the corresponding legs are congruent

(4) the areas are equal

10. In the diagram below, FE bisects AC at B, and ¿ bisects BD at C .

Which statement is always true?

(1) AB≅ DC (3) BD bisects ¿ at C

(2) FB≅ EB (4) AC bisects FE at B

11. Triangle ABC and triangle ¿ are graphed on the set of axes below.

Which sequence of transformations maps triangle ABC onto triangle ¿?

(1) a reflection over the x-axis followed by a reflection over the y- axis

(2) a 180 ° rotation about the origin followed by a reflection over the line y=x

(3) a 90 ° clockwise rotation about the origin followed by a reflection over the y-axis

(4) a translation 8 units to the right and 1 units up followed by a 90 ° counterclockwise rotation about the origin

12. Quadrilateral ABCD is graphed on the set of axes to the right. When ABCD is rotated 90 ° in a counterclockwise direction about the origin, its image is quadrilateral A' B'C' D'. Is distance preserved under this rotation, and which coordinates are correct for the given vertex?

(1) no and C ' (1 ,2 ) (3) yes and A' (6 ,2 )

(2) no and D' (2 ,4 ) (4) yes and B' (−3 ,4 )

13. Line segment EA is the perpendicular bisector of ZT , and ZE and TE are drawn.

Which conclusion can not be proven?

(1) EA bisects angle ZET

(2) Triangle EZT is equilateral

(3) EA is a median of triangle EZT

(4) Angle Z is congruent to angle T

14. Find the value of ∠n.

∠n=¿ __________

15. In regular hexagon ABCDEF shown below, AD, BE , and CF all intersect at G.

When ∆ ABG is reflected over BG and then rotated 180 ° about point G, ∆ ABG is mapped onto

(1) ∆ FEG (3) ∆CBG

(2) ∆ AFG (4) ∆ DEG

16. In the diagram below, AKS, NKC , AN , and SC are drawn such that AN ≅ SC .

Which statement is sufficient to prove ∆ KAN ≅ ∆ KSC by AAS?

(1) AS and NC bisect each other

(2) K is the midpoint of NC .

(3) AS⊥CN

(4) AN ∥ SC

17. In ∆ ABC , BD is the perpendicular bisector of ADC. Based upon this information, which statements below can be proven?

I. BD is a medianII. BD bisects ∠ ABCIII. ∆ ABC is isosceles

(1) I and II, only (3) II and III, only

(2) I and III, only (4) I, II, and II

18. After a counterclockwise rotation about point X , scalene triangle ABC maps onto ∆ RST , as shown in the diagram below.

Which statement must be true?

(1) ∠ A≅∠R (3) CB≅ TR

(2) ∠ A≅∠ S (4) CA ≅ TS

19. In the diagram below, AB∥¿, AE and BD intersect at C , m∠B=43 °, and m∠CEF=152° .

Which statement is true?

(1) m∠D=28 ° (3) m∠ ACD=71 °

(2) m∠ A=43° (4) m∠BCE=109 °

20. In the diagram below, line m is parallel to line n. Figure 2 is the image of Figure 1 after a reflection over line m. Figure 3 is the image of Figure 2 after a reflection over line n.

Which single transformation would carry Figure 1 onto Figure 3?

(1) a dilation (3) a reflection

(2) a rotation (4) a translation

21. The regular polygon is rotated about its center.

What is the minimum angle of rotation that will carry the figure onto itself?

(1) 60 ° (3) 72 °

(2) 360 ° (4) 180 °

22. In parallelogram ABCD shown below, the bisectors of ∠ ABC and ∠DCB meet at E, a point of AD. If m∠ A=70 °, determine and state m∠BEA .

23. In isosceles trapezoid QRST shown below, QR and TS are bases. If m∠Q=5 x+3 and m∠R=7 x−15, find x.

24. In the diagram of PQRS below, the diagonals PR and QS intersect at point T , PR=10, and QS=24. Determine and state the perimeter of PQRS.

25. As shown in the accompanying diagram, a rectangular gate has two diagonal supports. If m∠1=42, what is m∠2?

26. In the diagram below of parallelogram ROCK , m∠K=114° and m∠ROS=56 ° . What is m∠RSO?

(1) 56 ° (3) 58 °

(2) 48 ° (4) 66 °

27. A parallelogram must be a rectangle when its

(1) diagonals are perpendicular

(2) diagonals are congruent

(3) opposite sides are parallel

(4) opposite sides are congruent

28. If the diagonals of a quadrilateral do not bisect each other, then the quadrilateral could be a

(1) rectangle

(2) rhombus

(3) square

(4) trapezoid

29. Given three distinct quadrilaterals, a square, a rectangle, and a rhombus, which quadrilaterals must have perpendicular diagonals?

(1) the rhombus, only

(2) the rectangle and the square

(3) the rhombus and the square

(4) the rectangle, the rhombus, and the square

30. In the diagram of parallelogram FRED shown below, ED is extended to A, and AF is drawn such that AF≅ DF

If m∠R=124 °, what is m∠FDE?

(1) 124 ° (3) 56 °

(2) 68 ° (4) 112°

31. A parallelogram must be a rhombus if its diagonals

(1) are congruent

(2) bisect each other

(3) do not bisect its angles

(4) are perpendicular to each other

32. In parallelogram QRST shown below, diagonal TR is drawn, U and V are points on TS and QR, respectively, and UV intersects at TR at W .

If m∠ S=60 °, m∠ SRT=83 °, and m∠TWU=35 °, what is m∠WVQ?

(1) 37 ° (3) 72 °

(2) 60 ° (4) 83 °

33. Which set of statements would describe a parallelogram that can always be classified as a rhombus?

I. Diagonals are perpendicular bisectors of each otherII. Diagonals bisect the angles from which they are drawn.III. Diagonals form four congruent isosceles right triangles

(1) I and II (3) II and III

(2) I and III (4) I, II, and III

34. Complete a formal proof, proving the triangles congruent, and then proving corresponding parts congruent. Be sure to mark your diagram to support your answer.

Given: KM bisects JN ∠JKL ≅∠NML

Prove: KL≅ML

35. Complete a formal proof, proving triangles are congruent. Be sure to mark your diagram to support your answer.

Given: XJ=YK PX=PY

Prove: ∆ JPY ≅ ∆ KPX

36. As shown in the diagram below, CD is a median of ∆ ABC . Which statement is always true?

(1) AD≅ DB (3) ∠ ACD≅∠CDB

(2) AC=AD (4) ∠BCD≅∠ ACD

37. In the diagram below, the vertices of ∆≝¿ are the midpoints of the sides of equilateral triangle ABC, and the perimeter of ∆ ABC is 36 cm. What is the length, in centimeters, of EF?

(1) 6 (3) 18

(2) 12 (4) 4

38. In ∆ ABC , D is the midpoint of AB and E is the midpoint of BC. If AC=3 x−15 and DE=6, what is the value of x?

(1) 6 (3) 9

(2) 7 (4) 12

39. In the diagram below of ∆ ACE, medians AD, EB, and CF intersect at G. The length of FG is 12 cm. What is the length, in centimeters, of GC?

(1) 24

(2) 12

(3) 6

(4) 4

40In the diagram below of ∆ ABC , DE is a mid-segment of ∆ ABC , DE=7 , AB=10, and BC=13. Find the perimeter of ∆ ABC .

41. In ∆ ABC shown below, medians AD, BE , and CF intersect at point R. If CR=24 and RF=2 x−6, what is the value of x?

42. Using a compass and a straightedge, construct the median to side AC in ∆ ABC below. [Leave all construction marks]

43. Using only a ruler and a compass, construct the bisector of angle BAC in the accompanying diagram.

44. Using a compass and a straightedge, locate the midpoint of AB by construction. [Leave all construction marks]

45. Using a compass and a straightedge, construct a line perpendicular to line l through point P. [Leave all construction marks.

46. Using a compass and a straightedge, construct a line perpendicular to AB through point P. [Leave all construction marks]

47. Using a compass and straightedge, on the diagram below of RS, construct an equilateral triangle with RS as one side. [Leave all construction marks]

48. Construct a line parallel to line l through point P.

49. Given trapezoid JKLM with JK∥ML using a compass and a straightedge, construct the altitude from vertex J to ML. Leave all construction marks.

50. Given ∆ ABC with m∠B=62° and side AC extended to D, as shown below.

Which value of x makes AB≅ CB?

(1) 59 °

(2) 62 °

(3) 118°

(4) 121 °

51. Which point shown in the graph below is the image of point P after a counterclockwise rotation of 90 ° about the origin?

52. Triangle ABC is graphed on the set of axes. Graph and label ∆ A 'B 'C ' the image of ∆ ABC after a reflection over the line y=−1.

53. A sequence of transformations maps rectangle ABCD onto rectangle A' ' B' 'C ' 'D ' ' .

a. Graph rectangle ABCD whose coordinates are A (−8 ,6 ) ,B (−2 ,6 ) ,C (−2 ,4 ) , and D (−8 ,4 ) on the coordinate plane below.

b. On the same set of axes, graph rectangle A' B'C' D', the image of rectangle ABCD reflected over the x-axis.

c. On the same set of axes, graph rectangle A' ' B' 'C ' 'D ' ' , the image of rectangle A' B'C' D' rotated about the origin 90 °.

d. State the coordinates of the final image, rectangle A' ' B' 'C ' 'D ' ' .

54. The vertices of ∆ RST are R (−6 ,5 ) , S (−7 ,−2 ) , and T (−1 ,4 ). The image of ∆ RST after a reflection about the y-axis followed by T−2,3 is ∆ R' ' S ' 'T ' '. Graph and label the coordinates of ∆ R' ' S ' 'T ' '.

55. Quadrilaterals BIKE and GOLF are graphed on the set of axes below.

Describe a sequence of transformations that maps quadrilateral BIKE onto quadrilateral GOLF.